Estimation of the number of irregular foreigners in Poland using non-linear count regression models

Population size estimation requires access to unit-level data in order to correctly apply capture-recapture methods. Unfortunately, for reasons of confidentiality access to such data may be limited. To overcome this issue we apply and extend the hierarchical Poisson-Gamma model proposed by Zhang (2008), which initially was used to estimate the number of irregular foreigners in Norway. The model is an alternative to the current capture-recapture approach as it does not require linking multiple sources and is solely based on aggregated administrative data that include (1) the number of apprehended irregular foreigners, (2) the number of foreigners who faced criminal charges and (3) the number of foreigners registered in the central population register. The model explicitly assumes a relationship between the unauthorized and registered population, which is motivated by the interconnection between these two groups. This makes the estimation conditionally dependent on the size of regular population, provides interpretation with analogy to registered population and makes the estimated parameter more stable over time. In this paper, we modify the original idea to allow for covariates and flexible count distributions in order to estimate the number of irregular foreigners in Poland in 2019. We also propose a parametric bootstrap for estimating standard errors of estimates. Based on the extended model we conclude that in as of 31.03.2019 and 30.09.2019 around 15,000 and 20,000 foreigners and were residing in Poland without valid permits. This means that those apprehended by the Polish Border Guard account for around 15-20% of the total.